{
  "id": "0705.2620",
  "version": "v3",
  "published": "2007-05-18T00:50:56.000Z",
  "updated": "2008-07-07T09:18:21.000Z",
  "title": "Homotopy groups of Hom complexes of graphs",
  "authors": [
    "Anton Dochtermann"
  ],
  "comment": "20 pages, 6 figures, final version, to be published in J. Combin. Theory Ser. A",
  "categories": [
    "math.CO",
    "math.AT"
  ],
  "abstract": "The notion of $\\times$-homotopy from \\cite{DocHom} is investigated in the context of the category of pointed graphs. The main result is a long exact sequence that relates the higher homotopy groups of the space $\\Hom_*(G,H)$ with the homotopy groups of $\\Hom_*(G,H^I)$. Here $\\Hom_*(G,H)$ is a space which parametrizes pointed graph maps from $G$ to $H$ (a pointed version of the usual $\\Hom$ complex), and $H^I$ is the graph of based paths in $H$. As a corollary it is shown that $\\pi_i \\big(\\Hom_*(G,H) \\big) \\cong [G,\\Omega^i H]_{\\times}$, where $\\Omega H$ is the graph of based closed paths in $H$ and $[G,K]_{\\times}$ is the set of $\\times$-homotopy classes of pointed graph maps from $G$ to $K$. This is similar in spirit to the results of \\cite{BBLL}, where the authors seek a space whose homotopy groups encode a similarly defined homotopy theory for graphs. The categorical connections to those constructions are discussed.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2008-07-07T09:18:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C15",
      "55P15",
      "57M15"
    ],
    "keywords": [
      "hom complexes",
      "higher homotopy groups",
      "parametrizes pointed graph maps",
      "long exact sequence",
      "homotopy groups encode"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0705.2620D"
    }
  }
}